arXiv research

Study of helices and Bertrand curves with modified orthogonal frame.

Study on Mannheim curves in 3D space with modified frame.

Study spherical images of modified vector fields on a unit sphere.

Modified knotoids with framing and coframing for quantum invariants.

AGI modifies utility function to cooperate, conflicting with orthogonality thesis.

New method for curve comparison using iterated integrals and moving frames.

Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…

The article proves Kähler 4-manifolds can only be products of 2 surfaces.

Modified Laplacian connects to Yang-Mills instantons on manifolds.

In this paper we define the bi-orthogonal sectional curvature and we present two modified Yamabe invariants for compact 4-dimensional manifolds. In particular we obtained a relationship between one of these invariants and a Hopf conjecture.

A Clifford algebra model for M"obius geometry is presented. The notion of Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced, and the structure equations for adapted frames are derived. These equations are discretized and the geometry of the occuring discrete nets and sphere congruences is disc…

The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.

Paper proves rigidity of spherical ring patterns on surfaces.

Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.

Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.

Geodesic flows between hypersurfaces in Euclidean spaces using Lorentzian geometry.

In the present work we construct a lift of a metric $g$ on a 2-dimensional oriented Riemannian manifold $M$ to a metric $\hat{g}$ on the total space $P$ of the orthonormal frame bundle of $M$. We call this lift the \textit {Wagner lift}. Viktor Vladimirovich Wagner (1908 -1981) proposed a technique to extend a metric d…

We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width at most three, then the flow categories are move equivalent. The proc…

Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.

The radar experiment connects the geometry of spacetime with an observers measurement of spatial length. We investigate the radar experiment on Finsler spacetimes which leads to a general definition of radar orthogonality and radar length. The directions radar orthogonal to an observer form the spatial equal time surfa…

Proposes a method to add adversarial framing to images without changing them.

Scalar-tensor gravitation theories, such as the Brans-Dicke family of theories, are commonly partly described by a modified Einstein equation in which the Ricci tensor is replaced by the Bakry-Émery-Ricci tensor of a Lorentzian metric and scalar field. In physics this formulation is sometimes referred to as the "Jordan…

We develop a frame and dyad gauge-independent formalism for the calculus of variations of functionals involving spinorial objects. As part of this formalism we define a modified variation operator which absorbs frame and spin dyad gauge terms. This formalism is applicable to both the standard spacetime (i.e. SL(2,C)) 2…

The paper examines a modified Godbillon-Vey class for Reeb foliations and finds it non-trivial for some foliations.

Modified Wasserstein metric for Gaussian distributions, invariant to isometries.

FPCA optimizes fairness in target vectors' span.

Transitive consistency is an intrinsic property for collections of linear invertible transformations between Euclidean coordinate frames. In practice, when the transformations are estimated from data, this property is lacking. This work addresses the problem of synchronizing transformations that are not transitively co…

Researchers prove a method to upgrade Morse-Bott homology to stable homotopy invariants.

The paper shows how gradient flow on over-parametrized tensor decomposition behaves like deflation.

Modified $Θ$-invariant for non-vanishing cohomology groups.

Matching cells over time has long been the most difficult step in cell tracking. In this paper, we approach this problem by recasting it as a classification problem. We construct a feature set for each cell, and compute a feature difference vector between a cell in the current frame and a cell in a previous frame. Then…

This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_VY$, the bundle of vertically adapted linear frames over the bundle of field configurations $Y$. Specifically, the generalized field momentum obs…

Improved modeling of chaotic systems using time-delay embeddings and Frenet-Serret frame.

Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, I considered the definition of orthonormal basis in Minkowsk…

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an (n)-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infi…

Rectifying curves on hypercones are geodesics, characterized in higher dimensions.

The paper studies connectivity of Schur-Horn map images in real Grassmannians.

GAIT-prop derives a biologically plausible learning rule from backpropagation.

In the paper we study the algebroid A of the groupoid of partially invertible elements over the lattice of orthogonal projections of a $W^*$-algebra. In particular the complex analytic manifold structure of these objects is investigated. The expressions on the Lie brackets for A and related algebroids are given in nonc…

This handbook simplifies Grassmann manifold geometry for matrix-based algorithms.

The paper studies steady motions of fibre-reinforced fluids on curved surfaces.

It has long been known to mathematicians and physicists that while a full rotation in three-dimensional Euclidean space causes tangling, two rotations can be untangled. Formally, an untangling is a based nullhomotopy of the double-twist loop in the special orthogonal group of rotations. We study a particularly simple, …

Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger …

BLOCCS improves sparse CCA for better interpretation of multi-omics data.

New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows $\map(t,x)$ in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. The derivation of these soliton hierarch…

We propose a novel application of the Simultaneous Orthogonal Matching Pursuit (S-OMP) procedure for sparsistant variable selection in ultra-high dimensional multi-task regression problems. Screening of variables, as introduced in \cite{fan08sis}, is an efficient and highly scalable way to remove many irrelevant variab…

Enhances neural networks' explainability without sacrificing accuracy.